Finding Vertices, Edges, Faces, and Tris using Python
You can access the polygons of a mesh, then calculate the triangles from the number of vertices per polygon.

import bpy

obj = bpy.context.scene.objects.active
data = obj.data

# FACES, TRIANGLES
total_triangles = 0

for face in data.polygons:
    vertices = face.vertices
    triangles = len(vertices) - 2
    total_triangles += triangles

print(total_triangles)

# VERTICES
vertices = []
for vertex in data.vertices:
    pos = [vertex.co[0], vertex.co[1], vertex.co[2]]
    vertices.append(pos)

print(vertices)

# EDGES
edges = []
for edge in data.edges:
    edges.append([edge.vertices[0], edge.vertices[1]])

print(edges)